A catapult launches a boulder with an upward velocity of 112 ft/s. The height of the boulder,h, in feet after t seconds is given by the function h=-16t^2 + 112t + 30. How long does it take the boulder to reach its maximum height? What is the boulders maximum height? Round to the nearest hundredth,if necessary.

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anonymous · Answer

Calculus class?

anonymous · Answer

I gotta know. Can we use derivatives to solve this, or are we limited to analysis of the parabola?

anonymous · Answer

yea - i was puzzled by this question i - its easily by newtons  equations too  , if you neglect  air friction

anonymous · Answer

Well, I'll assume this is a calculus course for now. Let me know if it's not.

So the idea is that we have a function for the height of the projectile. If we take the derivative of that function, that will give us an equation for the rate of change of the height, that is, it'll tell us how quickly the projectile is going up.
The projectile will start to go up really quickly at first, but then it'll slow down and finally STOP for just an instant before dropping again. At this point, the projectile is at the highest it will get, and it's not going up or down. So the derivative isn't positive (upward) or negative (downward). It's just 0.

So! We can take the derivative of the height function and figure out when this derivative is 0. That tells us the time that the projectile is at its highest. Putting that time back into the height equation will tell us exactly how high it gets.